One reason why the fiscal cliff negotiations haven’t gotten very far, at least in public, is that Republicans are hung up on the matter of exactly what they’re trying to achieve. It’s not that they don’t know the answer — they want deep cuts to entitlement spending — but rather they don’t want to advertise it, since cutting Medicare, Medicaid, and Social Security is unpopular and therefore politically costly. But it’s hard to move forward if you’re not willing to state your demands, and so things have bogged down since President Obama opened the negotiations by asking for $1.6 trillion in tax increases and deduction caps.

For Republicans, entitlement cuts are a long-sought-after prize, but one that’s difficult to acknowledge openly. Over the years, Democrats have reliably pursued a strategy known as “Mediscare” — frightening seniors by dramatizing how such cuts would affect them. Last year, for instance, television ads attacking supporters of the Republican House budget famously showed a congressman shoving a wheelchair-bound granny off a cliff.

Those ads work. So when Republicans want to talk about cutting entitlements they tend to speak in code, declaring that we need to “get our spending under control” or “shrink government” rather than specify what programs would be cut or shrunk. In fact, they’re so frightened of a backlash that during the presidential campaign Mitt Romney and Paul Ryan attacked Obama from the left on Medicare, decrying the $716 billion in cuts enshrined in the health care law and vowing to restore them, despite the fact that Ryan’s own budget, passed by the House, would have imposed precisely the same cuts.

On the fiscal cliff, that’s put Republicans in a bind. They don’t have much leverage because without Democrats’ consent they cannot stop the tax increase on the top 2 percent of earners set to kick in at year’s end. Most Republicans realize, furthermore, that driving the country over the cliff and forcing everyone’s taxes to go up because Democrats won’t extend cuts for the rich — “holding the middle class hostage,” in Obama’s phrase — would be political suicide.

One idea that’s begun circulating among conservatives is that they might be better off accepting the inevitable tax hikes and instead building a strategy around what they presume to be their greatest point of leverage: the need to increase the debt limit early next year. The thinking here is that with the fiscal cliff behind them, they’ll be able to run the same play that they did in 2011, threatening default unless their demands are met. The White House wants to avoid this scenario. Its opening bid included ending Congress’s power over the debt limit, and Obama says he won’t agree to any deal that doesn’t raise it. He may end up getting one — but right now a showdown over the debt ceiling seems at least as likely as one over tax rates.

If such a showdown comes to pass, however, debt default is a threat whose power would almost certainly diminish as the deadline approaches.

That’s because, although a debt-limit fight wouldn’t pit the rich against the middle class in quite the same way as the fight over tax cuts, it would force Republicans to enumerate and defend precisely the entitlement cuts they are desperate to obscure. And they’d be doing so in a way guaranteed to maximize attention, because they’d be threatening to tank the economy — a far scarier prospect than going over the fiscal cliff.

Furthermore, it’s not at all clear that they would be any better off forcing entitlement cuts than they would be trying to force tax cuts for the rich. It’s true, as Republicans point out, that most people want to do something about the deficit. But polls show that Americans overwhelmingly favor raising taxes on the rich — and preserving programs like Medicare and Social Security.

For the time being, Republicans are ignoring this dissonance between their own desire and what the public is likely to accept. But as the default date looms, that tradeoff will become ever more explicit, and there’s no reason to imagine that it will become any more popular. Quite the opposite, in fact, which is why their best hope to cut entitlements is to negotiate for them now with a president more willing to concede them than most Republicans seem to understand.

Joshua Green is national correspondent for Bloomberg Businessweek. His column appears in The Boston Globe.